The lifespans of porcupines in a particular zoo are normally distributed. The average porcupine lives $19$ years; the standard deviation is $1.9$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a porcupine living less than $13.3$ years.
Solution: $19$ $17.1$ $20.9$ $15.2$ $22.8$ $13.3$ $24.7$ $99.7\%$ $0.15\%$ $0.15\%$ We know the lifespans are normally distributed with an average lifespan of $19$ years. We know the standard deviation is $1.9$ years, so one standard deviation below the mean is $17.1$ years and one standard deviation above the mean is $20.9$ years. Two standard deviations below the mean is $15.2$ years and two standard deviations above the mean is $22.8$ years. Three standard deviations below the mean is $13.3$ years and three standard deviations above the mean is $24.7$ years. We are interested in the probability of a porcupine living less than $13.3$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the porcupines will have lifespans within 3 standard deviations of the average lifespan. The remaining $0.3\%$ of the porcupines will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({0.15\%})$ will live less than $13.3$ years and the other half $({0.15\%})$ will live longer than $24.7$ years. The probability of a particular porcupine living less than $13.3$ years is ${0.15\%}$.